SO(4,C)-covariant Ashtekar-Barbero gravity and the Immirzi parameter

TL;DR

This work presents a covariant $SO(4,\mathbb{C})$ canonical formulation of a generalized Hilbert--Palatini action parameterized by the Immirzi parameter $\beta$, avoiding time gauge fixing and embedding Ashtekar--Barbero gravity within a unified so(4,$\mathbb{C}$) framework. It derives a covariant 3+1 decomposition, identifies first- and second-class constraints, and shows that the resulting Dirac brackets render the gauge algebra as the adjoint of $so(4,\mathbb{C})$, with a true connection $A_i^X$. In the quantum analysis, the path integral is shown to be formally independent of $\beta$ in the chosen gauge, while the standard loop quantization faces significant obstacles due to noncommutativity of the connection and the Lorentz group’s noncompactness. The work outlines potential directions for a Lorentz-covariant loop approach and signals possible connections to noncommutative geometry, highlighting both promise and substantial technical challenges for quantizing gravity in this covariant setting.

Abstract

An so(4,C)-covariant hamiltonian formulation of a family of generalized Hilbert-Palatini actions depending on a parameter (the so called Immirzi parameter) is developed. It encompasses the Ashtekar-Barbero gravity which serves as a basis of quantum loop gravity. Dirac quantization of this system is constructed. Next we study dependence of the quantum system on the Immirzi parameter. The path integral quantization shows no dependence on it. A way to modify the loop approach in the accordance with the formalism developed here is briefly outlined.